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先来预习一下。

Week 0基本概念一、样本空间、随机事件定义
随机试验
人们是通过试验去研究随机现象的，,"> 
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            <h1 id="info"><a href="#info" class="headerlink" title="info"></a>info</h1><p>明天就要上概率论与数理统计这门课了，做个笔记，B站上宋浩老师讲的真的快乐的一批。</p>
<p>先来预习一下。</p>
<hr>
<h1 id="Week-0"><a href="#Week-0" class="headerlink" title="Week 0"></a>Week 0</h1><h1 id="基本概念"><a href="#基本概念" class="headerlink" title="基本概念"></a>基本概念</h1><h2 id="一、样本空间、随机事件"><a href="#一、样本空间、随机事件" class="headerlink" title="一、样本空间、随机事件"></a>一、样本空间、随机事件</h2><h3 id="定义"><a href="#定义" class="headerlink" title="定义"></a>定义</h3><ol>
<li><p><strong>随机试验</strong></p>
<p>人们是通过试验去研究随机现象的，把对随机现象加以研究所进行的观察或实验，称为试验.若一个试验具有下列三个特点：</p>
<p>1°可以在相同的条件下重复地进行；</p>
<p>2°每次试验的可能结果不止一个，并且事先可以明确试验所有可能出现的结果；</p>
<p>3°进行一次试验之前不能确定哪一个结果会出现.</p>
<p>则称这一试验为随机试验（random trial），记为<code>E</code>.</p>
<pre><code class="txt">e.g.
E1：抛一枚硬币，观察正面H和反面T出现的情况；
E2：掷两颗骰子，观察出现的点数；
E3：在一批电视机中任意抽取一台，测试它的寿命；</code></pre>
</li>
<li><p><strong>样本空间与随机事件</strong></p>
<p>在一个试验中，不论可能出现的结果有多少，总可以从中找出一组基本结果，满足： </p>
<p>1°每进行一次试验，必然出现且只能出现其中的一个基本结果.</p>
<p>2°任何结果，都是由其中的一些基本结果所组成.</p>
<hr>
<ul>
<li><p>随机试验<code>E</code>的所有基本结果组成的集合称为样本空间（sample space），记为$\Omega$.样本空间的元素，即<code>E</code>的每个基本结果，称为<code>样本点</code>.</p>
</li>
<li><p>随机试验<code>E</code>的样本空间<code>Ω</code>的子集称为<code>E</code>的随机事件（random event），简称事件，通常用大写字母<em>A</em>，<em>B</em>，<em>C</em>，…表示.在每次试验中，当且仅当这一子集中的一个样本点出现时，称<code>这一事件发生</code>.</p>
</li>
<li><p>特别地，由一个样本点组成的单点集，称为基本事件.</p>
</li>
</ul>
</li>
<li><p><strong>事件之间的关系及其运算</strong></p>
</li>
</ol>
<pre><code class="txt">  1°&quot;如果事件`A`发生必然导致事件`B`发生&quot;，则称事件`A`包含于事件`B`（或称事件`B`包含事件`A`）

  2°&quot;事件`A`与`B`中至少有一个发生&quot;的事件称为`A`与`B`的并（和）。两两互斥的情况下，并可以用`+`表示.

  3°&quot;事件`A`与`B`同时发生&quot;的事件称为`A`与`B`的交（积）

  4°“事件A发生而B不发生”的事件称为A与B的差

  5°如果两个事件A与B不可能同时发生，则称事件A与B为互不相容（互斥）

  6°若A∪B=Ω且A∩B=空集，则称事件A与事件B互为逆事件（对立事件）.A的对立事件=Ω-A</code></pre>
<pre><code class="txt">   e.g.
   Ω1:{H,T}；
   Ω2:{(i,j)｜i,j=1,2,3,4,5,6}；
   Ω3:{t｜t≥0}</code></pre>
<p>   <img src="https://i.loli.net/2020/03/07/dBWSHRea1vJcMm5.png" alt="vn图.png"></p>
<ol start="4">
<li><p>运算律</p>
<p><img src="https://i.loli.net/2020/03/07/MqkQiwToc6ysKGf.png" alt="运算律.png"></p>
</li>
</ol>
<h3 id="例题"><a href="#例题" class="headerlink" title="例题"></a>例题</h3><p><img src="https://i.loli.net/2020/03/07/5MtevAqyR6pcZga.png" alt="例1-1.png"></p>
<hr>
<p><img src="https://i.loli.net/2020/03/07/LtKbMq4Cg7NcEIS.png" alt="例1-2.png"></p>
<hr>
<h3 id="other"><a href="#other" class="headerlink" title="other"></a>other</h3><ol>
<li>有限&amp;无限样本空间</li>
<li>同一个描述可以有多种表示方法</li>
<li>运算律</li>
</ol>
<hr>
<h2 id="二、概率-amp-古典概型"><a href="#二、概率-amp-古典概型" class="headerlink" title="二、概率&amp;古典概型"></a>二、概率&amp;古典概型</h2><h3 id="定义-1"><a href="#定义-1" class="headerlink" title="定义"></a>定义</h3><ol>
<li>Intro</li>
</ol>
<ul>
<li><p><strong>加法原理</strong></p>
<p>做一件事情，完成它有n类方式，第一类方式有M1种方法，第二类方式有M2种方法，……，第n类方式有Mn种方法，那么完成这件事情共有M1+M2+……+Mn种方法。</p>
</li>
<li><p><strong>乘法原理</strong></p>
<p>做一件事，完成它需要分成n个步骤，做第一步有M1种不同的方法，做第二步有M2种不同的方法，……，做第n步有Mn种不同的方法。那么完成这件事共有 N=M1×M2×M3×…×Mn 种不同的方法。</p>
</li>
<li><p><strong>排列&amp;组合</strong></p>
<ol>
<li><p>排列：<br>$$<br>{A_n^m}=n(n-1)(n-2)…(n-m+1)={n!\over {(n-m)!}}<br>$$</p>
</li>
<li><p>组合：<br>$$<br>\begin{align}<br>C(n,m)=C(n,n-m) \qquad(n\ge m)<br>\end{align}<br>$$</p>
<p><img src="https://gitee.com/L3SLIE/blog/raw/master/img/%E5%85%AC%E5%BC%8F.png" alt="公式"></p>
</li>
</ol>
</li>
</ul>
<ol start="2">
<li><p>频率</p>
<ul>
<li><p>定义</p>
<p><img src="https://i.loli.net/2020/03/07/uHUA1yF23VZpgTI.png" alt="频率定义.png"></p>
</li>
<li><p>性质</p>
<ol>
<li>非负性</li>
<li>规范性</li>
<li>有限可加性（A、B互不相容）</li>
</ol>
</li>
</ul>
</li>
<li><p>频率的稳定性</p>
<p>当试验次数n增大时，随机事件的频率$f_n(A)$逐渐趋向稳定</p>
</li>
<li><p>统计概率</p>
<ul>
<li><p>统计定义</p>
<p>设有随机试验，若当试验的次数充分大时，事件A发生的频率稳定在某数<code>p</code>附近摆动，则称数<code>p</code>为事件A发生的统计概率，记为：$P(A)=p$</p>
<ol>
<li>一个事件发生的概率完全决定于事件本身的结构，指试验条件，是先于试验而客观存在的。</li>
<li>概率的统计定义只是描述性的</li>
<li>通常只能在充分大时，事件出现的频率才作为事件概率的近似值</li>
</ol>
</li>
<li><p>概率的公理化定义</p>
<p>设随机试验的样本空间为Ω，若对每一事件A，有且只有一个实数<code>P(A)</code>与之对应，满足如下公理：</p>
<ol>
<li>非负性 $0≤P(A)≤ 1$</li>
<li>规范性 $P(Ω) = 1$</li>
<li>完全可加性（和的概率=概率的和）</li>
</ol>
</li>
<li><p>概率的性质</p>
<ol>
<li><p>$$<br>P(\phi)=0概率为0的事件为不可能事件<br>$$</p>
</li>
<li><p>有限可加性：若事件A和B<code>互不相容</code>，则有 $P(A+B)=P(A)+P(B)$</p>
</li>
<li><p>$$<br>若A\subset B，则有P(B-A)=P(B)-P(A)\且有P(B)\ge P(A)<br>$$</p>
</li>
<li><p>对任意事件A和B，有$P(B-A)=P(B-BA)=P(B)-P(BA)$</p>
</li>
<li><p>$$<br>对任意事件A，有P(A)=1-P(\overline A)<br>$$</p>
</li>
<li><p>$$<br>两事件A和B之和的概率等于其概率之和减去积AB的概率，\即P(A\cup B)=P(A)+P(B)-P(AB)<br>$$</p>
</li>
<li><p>加法公式：</p>
<p><img src="https://i.loli.net/2020/03/07/roq8yQPmaxdZpCk.png" alt="加法公式.png"></p>
</li>
</ol>
</li>
</ul>
</li>
</ol>
<h3 id="例题1-4"><a href="#例题1-4" class="headerlink" title="例题1-4"></a>例题1-4</h3><p>   <img src="https://i.loli.net/2020/03/07/hKGwWuNm5f4IjVr.png" alt="例1-4.png"></p>
<ol start="5">
<li><p>古典概型</p>
<ul>
<li><p>定义</p>
<p><img src="https://i.loli.net/2020/03/07/8DMLKX6C42HWBoR.png" alt="古典概型定义.png"></p>
</li>
<li><p>运算公式</p>
<p>设A有k个基本事件，<code>P(A)=k/n=A中样本总数/Ω中样本总数</code>.</p>
</li>
</ul>
</li>
</ol>
<h3 id="例题1-5"><a href="#例题1-5" class="headerlink" title="例题1-5"></a>例题1-5</h3><p><img src="https://i.loli.net/2020/03/07/mHgTe1yZDj5VtBf.png" alt="例1-5.png"></p>
<h3 id="例题1-6"><a href="#例题1-6" class="headerlink" title="例题1-6"></a>例题1-6</h3><p><img src="https://i.loli.net/2020/03/07/Ky6P2m1wl4EUoe9.png" alt="例1-6.png"></p>
<p><img src="https://i.loli.net/2020/03/07/wDqSZTIGbmEngAC.png" alt="1-6解_1.png"></p>
<p><img src="https://i.loli.net/2020/03/07/3TMszE1oSHvX8Nn.png" alt="1-6解_2.png"></p>
<hr>
<ol start="6">
<li><p>几何概型</p>
<p>上述古典概型的计算，只适用于具有等可能性的有限样本空间，若试验结果无穷多，它显然已不适合.为了克服有限的局限性，可将古典概型的计算加以推广</p>
<p><strong>1°</strong>样本空间Ω是一个几何区域，这个区域大小可以度量（如长度、面积、体积等），并把Ω的度量记作<code>m（Ω）</code>.</p>
<p><strong>2°</strong>向区域Ω内任意投掷一个点，落在区域内任一个点处都是“等可能的”.或者设落在Ω中的区域A内的可能性与A的度量<code>m(A)</code>成正比，与A的<code>位置和形状无关</code>.</p>
<p>不妨也用<em>A</em>表示“掷点落在区域<em>A</em>内”的事件，那么事件<em>A</em>的概率可用下列公式计算：</p>
<p><em>P</em>（<em>A</em>）=<em>m</em>(<em>A</em>)/<em>m</em>(Ω)，</p>
<p>称它为几何概率.</p>
</li>
</ol>
<h3 id="例题1-10"><a href="#例题1-10" class="headerlink" title="例题1-10"></a>例题1-10</h3><p><img src="https://i.loli.net/2020/03/08/fHXsRMYoEm4W7rI.png" alt="例1-10.png"></p>
<h3 id="例题1-11"><a href="#例题1-11" class="headerlink" title="例题1-11"></a>例题1-11</h3><p><img src="https://i.loli.net/2020/03/08/SuNivgAkLHTOYzy.png" alt="例题1-11.png"></p>
<h1 id="Week-1"><a href="#Week-1" class="headerlink" title="Week 1"></a>Week 1</h1><h2 id="三、条件概率、全概率公式"><a href="#三、条件概率、全概率公式" class="headerlink" title="三、条件概率、全概率公式"></a>三、条件概率、全概率公式</h2><h3 id="定义-2"><a href="#定义-2" class="headerlink" title="定义"></a>定义</h3><ol>
<li><p>条件概率</p>
<p><strong>定义1.5</strong>    设<em>A</em>，<em>B</em>为两个事件，且<em>P</em>（<em>B</em>）＞0,则称<em>P</em>（<em>AB<em>）/</em>P<em>（</em>B<em>）为事件</em>B<em>已发生的条件下事件</em>A<em>发生的条件概率，记为</em>P<em>（</em>A</em>|<em>B</em>），即</p>
<p><code>P（A｜B)= P(AB)/P(B)</code></p>
<p>易验证，<em>P</em>（<em>A</em>｜<em>B</em>)符合概率定义的3条公理，即：</p>
<p><img src="https://i.loli.net/2020/03/08/Co7LmahNvPylF94.png" alt="定义1.5.png"></p>
<p>其中<em>A</em>1，<em>A</em>2，…，<em>An</em>，…为两两互不相容事件.</p>
<p><img src="https://i.loli.net/2020/03/08/TpEUl7yXFdQY28s.png" alt="定义1.5.1.png"></p>
</li>
</ol>
<h3 id="例题1-13"><a href="#例题1-13" class="headerlink" title="例题1-13"></a>例题1-13</h3><p><img src="https://i.loli.net/2020/03/08/M5bS1o7pJHqKuFa.png" alt="例题1-13.png"></p>
<h3 id="例题1-14"><a href="#例题1-14" class="headerlink" title="例题1-14"></a>例题1-14</h3><p><img src="https://i.loli.net/2020/03/08/qCOGSpkxeV349IT.png" alt="例题1-14.png"></p>
<ol start="2">
<li><p>乘法定理</p>
<p>由条件概率定义<em>P</em>（<em>B</em>｜<em>A</em>）=<em>P</em>（<em>AB</em>）/<em>P</em>（<em>A</em>）（<em>P</em>（<em>A</em>）＞0）,两边同乘以<em>P</em>（<em>A</em>）可得</p>
<p><em>P</em>（<em>AB</em>）=<em>P</em>（<em>A</em>）<em>P</em>（<em>B</em>｜<em>A</em>），由此可得下述定理：</p>
<p><img src="https://i.loli.net/2020/03/08/hx3RUBHKMCjysPn.png" alt="乘法定理.png"></p>
</li>
</ol>
<h3 id="例题1-15-16"><a href="#例题1-15-16" class="headerlink" title="例题1-15/16"></a>例题1-15/16</h3><p><img src="https://i.loli.net/2020/03/08/FOQK23EaHYymw7q.png" alt="例题1-15.16.png"></p>
<ol start="3">
<li><p>全概率公式&amp;贝叶斯公式</p>
<p>为建立两个用来计算概率的重要公式，我们先引入样本空间Ω的划分的定义.</p>
<p><strong>定义1.6</strong>    设Ω为样本空间，<em>A</em>1，<em>A</em>2，…，<em>An</em>为Ω的一组事件，若满足<br>$$<br>\begin{align}<br>&amp;1.A_iA_j=\Phi (i\neq j,i,j=1,2,3…,n)\<br>&amp;2.A_1\cup A_2\cup …\cup A_n=\Omega<br>\end{align}<br>$$<br>则称<em>A</em>1，<em>A</em>2，…，<em>An</em>为样本空间Ω的一个划分.</p>
<p>全概率公式：</p>
<p><img src="https://i.loli.net/2020/03/08/eW4AHViPNrTp9U7.png" alt="全概率公式.png"></p>
<p>贝叶斯公式：</p>
<p><img src="https://i.loli.net/2020/03/08/OhsrHw4EBFjtQUL.png" alt="贝叶斯公式.png"></p>
</li>
</ol>
<h3 id="例题1-18-19-20"><a href="#例题1-18-19-20" class="headerlink" title="例题1-18/19/20"></a>例题1-18/19/20</h3><p><img src="https://i.loli.net/2020/03/08/Mg8GcldSjyRQ4Ce.png" alt="例题1-18.png"></p>
<p><img src="https://i.loli.net/2020/03/08/rTlLm2e9OA5kVZx.png" alt="例题1-19.png"></p>
<p><img src="https://i.loli.net/2020/03/08/zbaiZyYHfSAcrFe.png" alt="例题1-20.png"></p>
<p>这就是说，根据以往的数据分析可以得到，患有癌症的被诊断者，试验反应为阳性的概率为95%，没有患癌症的被诊断者，试验反应为阴性的概率为95%，都叫做先验概率.而在得到试验结果反应为阳性，该被诊断者确有癌症的重新加以修正的概率0.087叫做后验概率.此项试验也表明，用它作为普查，正确性诊断只有8.7%（即1000人具有阳性反应的人中大约只有87人的确患有癌症），由此可看出，若把<em>P</em>（<em>B</em>｜<em>A</em>）和<em>P</em>（<em>A</em>｜<em>B</em>）搞混淆就会造成误诊的不良后果.</p>
<p>概率乘法公式、全概率公式、贝叶斯公式称为条件概率的3个重要公式.它们在解决某些复杂事件的概率问题中起到十分重要的作用.</p>
<h2 id="四、独立性"><a href="#四、独立性" class="headerlink" title="四、独立性"></a>四、独立性</h2><h3 id="定义-3"><a href="#定义-3" class="headerlink" title="定义"></a>定义</h3><ol>
<li><p>事件的独立性</p>
<p>独立性是概率统计中的一个重要概念，在讲独立性的概念之前先介绍一个例题.</p>
<p><code>定义1.7</code>    若事件<em>A</em>1，<em>A</em>2满足</p>
<p><em>P</em>（<em>A</em>1<em>A</em>2）=<em>P</em>（<em>A</em>1）<em>P</em>（<em>A</em>2），</p>
<p>则称事件<em>A</em>1，<em>A</em>2是相互独立的.</p>
<p><code>定理1.4</code>    若事件A与B相互独立，则A反与B反，A与B反，A反与B都相互独立</p>
<p><img src="https://i.loli.net/2020/03/08/WSEtXaG2KMFmR3Z.png" alt="定理1.4证明.png"></p>
<p><code>定理1.5</code>    </p>
<p><img src="https://i.loli.net/2020/03/08/BUbVmCYAIGhdXzH.png" alt="定理1.5.png"></p>
<p><code>定义1.9</code>   </p>
<p><img src="https://i.loli.net/2020/03/08/Hlp7RMG6uvNxyW2.png" alt="定义1.9.png"></p>
</li>
<li><p>伯努利实验</p>
<p>随机现象的规律性要从大量的现象中分析得出.在相同条件下进行重复试验或观察，是一种非常重要的概率模型.</p>
<p><img src="https://i.loli.net/2020/03/08/fG9Rt8o3OET5H7e.png" alt="n重伯努利实验.png"></p>
</li>
</ol>
<h3 id="例题-1"><a href="#例题-1" class="headerlink" title="例题"></a>例题</h3><p><img src="https://s2.ax1x.com/2020/03/08/3vjZ80.png" alt="3vjZ80.png"></p>
<p><img src="https://s2.ax1x.com/2020/03/08/3vjFEj.png" alt="3vjFEj.png"></p>
<p><img src="https://s2.ax1x.com/2020/03/08/3vjVCq.png" alt="3vjVCq.png"></p>
<p><img src="https://s2.ax1x.com/2020/03/08/3vjkUs.png" alt="3vjkUs.png"></p>
<p><img src="https://s2.ax1x.com/2020/03/08/3vjA5n.png" alt="3vjA5n.png"></p>
<h1 id="Summary"><a href="#Summary" class="headerlink" title="Summary"></a>Summary</h1><p>Key word：</p>
<p>随机试验  样本空间  随机事件</p>
<p>基本事件  频率     概率</p>
<p>古典概型  <em>A</em>的对立事件A反及其概率</p>
<p>两个互不相容事件的和事件的概率  概率的加法公式</p>
<p>条件概率  概率的乘法公式  全概率公式</p>
<p>贝叶斯公式 事件的独立性   <em>n</em>重伯努利试验</p>

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                    <ol class="toc"><li class="toc-item toc-level-1"><a class="toc-link" href="#info"><span class="toc-number">1.</span> <span class="toc-text">info</span></a></li><li class="toc-item toc-level-1"><a class="toc-link" href="#Week-0"><span class="toc-number">2.</span> <span class="toc-text">Week 0</span></a></li><li class="toc-item toc-level-1"><a class="toc-link" href="#基本概念"><span class="toc-number">3.</span> <span class="toc-text">基本概念</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#一、样本空间、随机事件"><span class="toc-number">3.1.</span> <span class="toc-text">一、样本空间、随机事件</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#定义"><span class="toc-number">3.1.1.</span> <span class="toc-text">定义</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#例题"><span class="toc-number">3.1.2.</span> <span class="toc-text">例题</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#other"><span class="toc-number">3.1.3.</span> <span class="toc-text">other</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#二、概率-amp-古典概型"><span class="toc-number">3.2.</span> <span class="toc-text">二、概率&amp;古典概型</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#定义-1"><span class="toc-number">3.2.1.</span> <span class="toc-text">定义</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#例题1-4"><span class="toc-number">3.2.2.</span> <span class="toc-text">例题1-4</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#例题1-5"><span class="toc-number">3.2.3.</span> <span class="toc-text">例题1-5</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#例题1-6"><span class="toc-number">3.2.4.</span> <span class="toc-text">例题1-6</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#例题1-10"><span class="toc-number">3.2.5.</span> <span class="toc-text">例题1-10</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#例题1-11"><span class="toc-number">3.2.6.</span> <span class="toc-text">例题1-11</span></a></li></ol></li></ol></li><li class="toc-item toc-level-1"><a class="toc-link" href="#Week-1"><span class="toc-number">4.</span> <span class="toc-text">Week 1</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#三、条件概率、全概率公式"><span class="toc-number">4.1.</span> <span class="toc-text">三、条件概率、全概率公式</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#定义-2"><span class="toc-number">4.1.1.</span> <span class="toc-text">定义</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#例题1-13"><span class="toc-number">4.1.2.</span> <span class="toc-text">例题1-13</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#例题1-14"><span class="toc-number">4.1.3.</span> <span class="toc-text">例题1-14</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#例题1-15-16"><span class="toc-number">4.1.4.</span> <span class="toc-text">例题1-15&#x2F;16</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#例题1-18-19-20"><span class="toc-number">4.1.5.</span> <span class="toc-text">例题1-18&#x2F;19&#x2F;20</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#四、独立性"><span class="toc-number">4.2.</span> <span class="toc-text">四、独立性</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#定义-3"><span class="toc-number">4.2.1.</span> <span class="toc-text">定义</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#例题-1"><span class="toc-number">4.2.2.</span> <span class="toc-text">例题</span></a></li></ol></li></ol></li><li class="toc-item toc-level-1"><a class="toc-link" href="#Summary"><span class="toc-number">5.</span> <span class="toc-text">Summary</span></a></li></ol>
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            valine.init({
                el: '#vcomments',
                appId: 'GnPlbHQ9xAWEoCDC8rR3zALj-gzGzoHsz',
                appKey: 'Nygd2awSxr9ioTDtvbH6hE0l',
                notify: 'false',
                verify: 'false',
                avatar: 'mm',
                placeholder: 'feel free to comment~',
                visitor: 'true',
                path: path,
                lang: language
            });
        }

        $(document).ready(function () {
            initValine();
        });
    </script>



</html>
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<script type="text/javascript" src="/js/love.js"></script>
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<canvas class="fireworks" style="position: fixed;left: 0;top: 0;z-index: 1; pointer-events: none;" ></canvas> 
<script type="text/javascript" src="//cdn.bootcss.com/animejs/2.2.0/anime.min.js"></script> 
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